#7278059

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,6)(3,4)(17,18,21,26)(19,22,27,24)(20,23,29,25)(28,31,32,30);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,21,27)(18,22,26,24)(20,28,29,32)(23,31,25,30);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(18,23)(19,27)(20,29)(22,30)(24,31)(25,26);(2,6)(7,8)(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,28)(21,27)(22,26)(23,30)(25,31)(29,32);(1,4,5,3)(2,8,6,7)(10,13)(14,16)(19,27)(22,24)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7185255);